Publication

Transport, Flux And Growth Of Homoclinic Floer Homology

2012
Journal Articles

Abstract

We point out an interesting relation between transport in Hamiltonian dynamics and Floer homology. We generalize homoclinic Floer homology from R-2 and closed surfaces to two-dimensional cylinders. The relative symplectic action of two homoclinic points is identified with the flux through a turnstile (as defined in MacKay & Meiss & Percival [19]) and Mather's [20] difference in action Delta W. The Floer boundary operator is shown to annihilate turnstiles and we prove that the rank of certain filtered homology groups and the flux grow linearly with the number of iterations of the underlying symplectomorphism.

Official source

https://infoscience.epfl.ch/entities/publication/5df6ae6a-cb77-4aba-a0ac-b87c8c87992d

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2012
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/5df6ae6a-cb77-4aba-a0ac-b87c8c87992d

About this result

Related concepts (22)

Floer homology

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold.

Khovanov homology

In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov, then at the University of California, Davis, now at Columbia University. To any link diagram D representing a link L, we assign the Khovanov bracket [D], a cochain complex of graded vector spaces. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial.

Persistent homology

See homology for an introduction to the notation. Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters. To find the persistent homology of a space, the space must first be represented as a simplicial complex.

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